Scalar Curvature, Injectivity Radius and Immersions with Small Second Fundamental Forms

Abstract

We prove special cases of the following.

●_Sc Bounds on the injectivity radii of “topologically complicated” Riemannian n-manifolds X , where the scalar curvatures of X are bounded from below, Sc(X ) ≥ σ > 0.

●_curv Lower bounds on focal radii of smooth immersions from k-manifolds, e.g. homeo- morphic to the k-torus, to certain Riemannian manifolds of dimensions n = k + m, e.g. to the cylinders S^{n−1} × R^1.

●_mean Topological lower bounds on the mean curvatures of domains in Riemannian manifolds. e.g. in the Euclidean n-space R^n.

At the present moment, our results are mainly limited by the spin condition and the n ≤ 8 restriction with additional difficulties in the case of foliations. The corresponding problems and questions are presented in shades of red.